ASSIGNMENT 8

Investigation of the relationship between the angles of a given triangle and the triangle formed by the intersections of the angle bisectors with the circumcircle

by

S. Kastberg


The internal angle bisectors of triangle ABC are extended to meet the circumcircle at points L,M, and N, respectively. Find the angles of triangle LMN in terms of angles A,B, and C.

The construction used to investigate this problem in Geometer's Sketchpad is not too difficult, however finding a relationship between angle M and angles A,B,C is a bit more challenging. Can you guess the relationship between M, A, C by looking at the the following diagram? If you would like to investigate this problem using GSP click HERE.

To find the measure of angle M consider the diagram above. Note that the measure of angle M = 1/2 the measure of angle A + 1/2 the measure of angle C. This can be proved by examining the measures of the angles subtended by the arcs indicated in red and the measures of the angles subtended by the arcs indicated in black. Notice that angle M is subtended by arc NBL, the sum of the red and black arcs. Now the formal proof. Please note that any reference to < XYZ is the measure of < XYZ.

Proof:

< NML = < NMB + < BML but, < BAL = < BML and < NMB = < NCB since, inscribled angles subtended by the same arc have equal measures. Since segment AL bisects < A and similarly segment CN bisects < C, < BAL = 1/2 < A and < NCB = 1/2 < C . Hence using appropriate substitutions < NML = 1/2 < A + 1/2 < C. QED.

A similar proof can be used to show that:

measure of angle N = 1/2 the measure of angle A + 1/2 the measure of angle B and

measure of angle L = 1/2 the measure of angle B + 1/2 the measure of angle C.

To investigate the relationship described above for angle N click HERE.

To invesigate the relationship described above for angle L click HERE.


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